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Theorem ce2le 6233
 Description: Partial ordering law for base two cardinal exponentiation. Theorem 4.8 of [Specker] p. 973. (Contributed by SF, 16-Mar-2015.)
Assertion
Ref Expression
ce2le (((M NC N NC (Nc 0c) NC ) Mc N) → (2cc M) ≤c (2cc N))

Proof of Theorem ce2le
Dummy variables y p q r s x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ce0t 6232 . . . 4 ((N NC (Nc 0c) NC ) → p NC N = Tc p)
213adant1 973 . . 3 ((M NC N NC (Nc 0c) NC ) → p NC N = Tc p)
32adantr 451 . 2 (((M NC N NC (Nc 0c) NC ) Mc N) → p NC N = Tc p)
4 letc 6231 . . . . . . . . 9 ((M NC p NC Mc Tc p) → q NC M = Tc q)
5 tlecg 6230 . . . . . . . . . . . . . . . . 17 ((q NC p NC ) → (qc pTc qc Tc p))
65ancoms 439 . . . . . . . . . . . . . . . 16 ((p NC q NC ) → (qc pTc qc Tc p))
7 elncs 6119 . . . . . . . . . . . . . . . . . . . 20 (q NCx q = Nc x)
8 elncs 6119 . . . . . . . . . . . . . . . . . . . 20 (p NCy p = Nc y)
97, 8anbi12i 678 . . . . . . . . . . . . . . . . . . 19 ((q NC p NC ) ↔ (x q = Nc x y p = Nc y))
10 eeanv 1913 . . . . . . . . . . . . . . . . . . 19 (xy(q = Nc x p = Nc y) ↔ (x q = Nc x y p = Nc y))
119, 10bitr4i 243 . . . . . . . . . . . . . . . . . 18 ((q NC p NC ) ↔ xy(q = Nc x p = Nc y))
12 enpw 6087 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (pxpx)
13 elnc 6125 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (p Nc xpx)
14 elnc 6125 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (p Nc xpx)
1512, 13, 143imtr4i 257 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p Nc xp Nc x)
1615adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((p Nc x q Nc y) → p Nc x)
1716adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((p Nc x q Nc y) p q) → p Nc x)
18 enpw 6087 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (qyqy)
19 elnc 6125 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (q Nc yqy)
20 elnc 6125 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (q Nc yqy)
2118, 19, 203imtr4i 257 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (q Nc yq Nc y)
2221adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((p Nc x q Nc y) → q Nc y)
2322adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((p Nc x q Nc y) p q) → q Nc y)
24 sspwb 4118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p qp q)
2524biimpi 186 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (p qp q)
2625adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((p Nc x q Nc y) p q) → p q)
27 sseq1 3292 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (r = p → (r sp s))
28 sseq2 3293 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (s = q → (p sp q))
2927, 28rspc2ev 2963 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((p Nc x q Nc y p q) → r Nc xs Nc yr s)
3017, 23, 26, 29syl3anc 1182 . . . . . . . . . . . . . . . . . . . . . . . 24 (((p Nc x q Nc y) p q) → r Nc xs Nc yr s)
3130ex 423 . . . . . . . . . . . . . . . . . . . . . . 23 ((p Nc x q Nc y) → (p qr Nc xs Nc yr s))
3231rexlimivv 2743 . . . . . . . . . . . . . . . . . . . . . 22 (p Nc xq Nc yp qr Nc xs Nc yr s)
33 ncex 6117 . . . . . . . . . . . . . . . . . . . . . . 23 Nc x V
34 ncex 6117 . . . . . . . . . . . . . . . . . . . . . . 23 Nc y V
3533, 34brlec 6113 . . . . . . . . . . . . . . . . . . . . . 22 ( Nc xc Nc yp Nc xq Nc yp q)
36 ncex 6117 . . . . . . . . . . . . . . . . . . . . . . 23 Nc x V
37 ncex 6117 . . . . . . . . . . . . . . . . . . . . . . 23 Nc y V
3836, 37brlec 6113 . . . . . . . . . . . . . . . . . . . . . 22 ( Nc xc Nc yr Nc xs Nc yr s)
3932, 35, 383imtr4i 257 . . . . . . . . . . . . . . . . . . . . 21 ( Nc xc Nc yNc xc Nc y)
40 vex 2862 . . . . . . . . . . . . . . . . . . . . . . 23 x V
4140tcnc 6225 . . . . . . . . . . . . . . . . . . . . . 22 Tc Nc x = Nc 1x
4240ce2 6192 . . . . . . . . . . . . . . . . . . . . . 22 ( Tc Nc x = Nc 1x → (2cc Tc Nc x) = Nc x)
4341, 42ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (2cc Tc Nc x) = Nc x
44 vex 2862 . . . . . . . . . . . . . . . . . . . . . . 23 y V
4544tcnc 6225 . . . . . . . . . . . . . . . . . . . . . 22 Tc Nc y = Nc 1y
4644ce2 6192 . . . . . . . . . . . . . . . . . . . . . 22 ( Tc Nc y = Nc 1y → (2cc Tc Nc y) = Nc y)
4745, 46ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (2cc Tc Nc y) = Nc y
4839, 43, 473brtr4g 4671 . . . . . . . . . . . . . . . . . . . 20 ( Nc xc Nc y → (2cc Tc Nc x) ≤c (2cc Tc Nc y))
49 breq12 4644 . . . . . . . . . . . . . . . . . . . . 21 ((q = Nc x p = Nc y) → (qc pNc xc Nc y))
50 tceq 6158 . . . . . . . . . . . . . . . . . . . . . . 23 (q = Nc xTc q = Tc Nc x)
5150oveq2d 5538 . . . . . . . . . . . . . . . . . . . . . 22 (q = Nc x → (2cc Tc q) = (2cc Tc Nc x))
52 tceq 6158 . . . . . . . . . . . . . . . . . . . . . . 23 (p = Nc yTc p = Tc Nc y)
5352oveq2d 5538 . . . . . . . . . . . . . . . . . . . . . 22 (p = Nc y → (2cc Tc p) = (2cc Tc Nc y))
5451, 53breqan12d 4654 . . . . . . . . . . . . . . . . . . . . 21 ((q = Nc x p = Nc y) → ((2cc Tc q) ≤c (2cc Tc p) ↔ (2cc Tc Nc x) ≤c (2cc Tc Nc y)))
5549, 54imbi12d 311 . . . . . . . . . . . . . . . . . . . 20 ((q = Nc x p = Nc y) → ((qc p → (2cc Tc q) ≤c (2cc Tc p)) ↔ ( Nc xc Nc y → (2cc Tc Nc x) ≤c (2cc Tc Nc y))))
5648, 55mpbiri 224 . . . . . . . . . . . . . . . . . . 19 ((q = Nc x p = Nc y) → (qc p → (2cc Tc q) ≤c (2cc Tc p)))
5756exlimivv 1635 . . . . . . . . . . . . . . . . . 18 (xy(q = Nc x p = Nc y) → (qc p → (2cc Tc q) ≤c (2cc Tc p)))
5811, 57sylbi 187 . . . . . . . . . . . . . . . . 17 ((q NC p NC ) → (qc p → (2cc Tc q) ≤c (2cc Tc p)))
5958ancoms 439 . . . . . . . . . . . . . . . 16 ((p NC q NC ) → (qc p → (2cc Tc q) ≤c (2cc Tc p)))
606, 59sylbird 226 . . . . . . . . . . . . . . 15 ((p NC q NC ) → ( Tc qc Tc p → (2cc Tc q) ≤c (2cc Tc p)))
6160imp 418 . . . . . . . . . . . . . 14 (((p NC q NC ) Tc qc Tc p) → (2cc Tc q) ≤c (2cc Tc p))
6261an32s 779 . . . . . . . . . . . . 13 (((p NC Tc qc Tc p) q NC ) → (2cc Tc q) ≤c (2cc Tc p))
63 breq1 4642 . . . . . . . . . . . . . . . 16 (M = Tc q → (Mc Tc pTc qc Tc p))
6463anbi2d 684 . . . . . . . . . . . . . . 15 (M = Tc q → ((p NC Mc Tc p) ↔ (p NC Tc qc Tc p)))
6564anbi1d 685 . . . . . . . . . . . . . 14 (M = Tc q → (((p NC Mc Tc p) q NC ) ↔ ((p NC Tc qc Tc p) q NC )))
66 oveq2 5531 . . . . . . . . . . . . . . 15 (M = Tc q → (2cc M) = (2cc Tc q))
6766breq1d 4649 . . . . . . . . . . . . . 14 (M = Tc q → ((2cc M) ≤c (2cc Tc p) ↔ (2cc Tc q) ≤c (2cc Tc p)))
6865, 67imbi12d 311 . . . . . . . . . . . . 13 (M = Tc q → ((((p NC Mc Tc p) q NC ) → (2cc M) ≤c (2cc Tc p)) ↔ (((p NC Tc qc Tc p) q NC ) → (2cc Tc q) ≤c (2cc Tc p))))
6962, 68mpbiri 224 . . . . . . . . . . . 12 (M = Tc q → (((p NC Mc Tc p) q NC ) → (2cc M) ≤c (2cc Tc p)))
7069com12 27 . . . . . . . . . . 11 (((p NC Mc Tc p) q NC ) → (M = Tc q → (2cc M) ≤c (2cc Tc p)))
7170rexlimdva 2738 . . . . . . . . . 10 ((p NC Mc Tc p) → (q NC M = Tc q → (2cc M) ≤c (2cc Tc p)))
72713adant1 973 . . . . . . . . 9 ((M NC p NC Mc Tc p) → (q NC M = Tc q → (2cc M) ≤c (2cc Tc p)))
734, 72mpd 14 . . . . . . . 8 ((M NC p NC Mc Tc p) → (2cc M) ≤c (2cc Tc p))
74733expa 1151 . . . . . . 7 (((M NC p NC ) Mc Tc p) → (2cc M) ≤c (2cc Tc p))
7574an32s 779 . . . . . 6 (((M NC Mc Tc p) p NC ) → (2cc M) ≤c (2cc Tc p))
76 breq2 4643 . . . . . . . . 9 (N = Tc p → (Mc NMc Tc p))
7776anbi2d 684 . . . . . . . 8 (N = Tc p → ((M NC Mc N) ↔ (M NC Mc Tc p)))
7877anbi1d 685 . . . . . . 7 (N = Tc p → (((M NC Mc N) p NC ) ↔ ((M NC Mc Tc p) p NC )))
79 oveq2 5531 . . . . . . . 8 (N = Tc p → (2cc N) = (2cc Tc p))
8079breq2d 4651 . . . . . . 7 (N = Tc p → ((2cc M) ≤c (2cc N) ↔ (2cc M) ≤c (2cc Tc p)))
8178, 80imbi12d 311 . . . . . 6 (N = Tc p → ((((M NC Mc N) p NC ) → (2cc M) ≤c (2cc N)) ↔ (((M NC Mc Tc p) p NC ) → (2cc M) ≤c (2cc Tc p))))
8275, 81mpbiri 224 . . . . 5 (N = Tc p → (((M NC Mc N) p NC ) → (2cc M) ≤c (2cc N)))
8382com12 27 . . . 4 (((M NC Mc N) p NC ) → (N = Tc p → (2cc M) ≤c (2cc N)))
8483rexlimdva 2738 . . 3 ((M NC Mc N) → (p NC N = Tc p → (2cc M) ≤c (2cc N)))
85843ad2antl1 1117 . 2 (((M NC N NC (Nc 0c) NC ) Mc N) → (p NC N = Tc p → (2cc M) ≤c (2cc N)))
863, 85mpd 14 1 (((M NC N NC (Nc 0c) NC ) Mc N) → (2cc M) ≤c (2cc N))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ⊆ wss 3257  ℘cpw 3722  ℘1cpw1 4135  0cc0c 4374   class class class wbr 4639  (class class class)co 5525   ≈ cen 6028   NC cncs 6088   ≤c clec 6089   Nc cnc 6091   Tc ctc 6093  2cc2c 6094   ↑c cce 6096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101  df-tc 6103  df-2c 6104  df-ce 6106 This theorem is referenced by:  nchoicelem9  6297
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